If I change the name, but log in with the old name, will the xmpp still work?
Yes, the "accounts" in the database are just to allow you to have more than one list of characters, since the client is hardcoded to 48 characters max. They have no relation to xmpp accounts other than by default it tries to use the same name to keep things consistent.
As an example of the compete disconnect there, Paragon Chat has a setting called 'offline mode' in the options panel which causes it to not use an XMPP connection at all. In that case you use the COH login screen to select which account from the database to use.
Also, where is the sqlite file kept?
[ParagonChat]\Database\ParagonChat.db
Also, I wanted to use a name with an underscore, but titan says I can't use those, only spaces, but if I do, will a name with a space work ingame (I'm guessing no).
XMPP doesn't allow spaces in account names, but COH did allow them in globals, so Paragon Chat maps spaces to underscores. It does that rather than the reverse in order to be consistent with Titan accounts, which in turn work that way because of the Wiki. They'll always show up as spaces in-game, IIRC.
The local account name in the database doesn't ever show up to other players and doesn't affect anything.
I set up some SQLite tools and tried opening it, but it wants the DB key and page size. I assumed you meant I could try it if I wanted to, but if giving out that information is not kosher, I understand.
I have no idea what that is asking. You shouldn't need any special settings to open the file. I just use sqlite3's command line tool and point it at the file. I know some people use a program called SQLite Explorer or something similar.
If you want to reduce the risk of breaking things while database editing, make sure data integrity is turned on. The SQL command for it is "PRAGMA foreign_keys=ON;" -- GUI tools may or may not have a checkbox somewhere to enable it.